The smallest semicopula-based universal integrals II: Convergence theorems

In this paper we continue studying the smallest universal integral IS having S as the underlying semicopula. We present convergence theorems for IS-integral sequences including monotone, almost everywhere, almost uniform, in measure and in mean converging sequences of measurable functions, respectively. It emerges that these convergences characterize the underlying measure properties such as null-additivity, monotone autocontinuity and autocontinuity. We provide many examples and counterexamples as well as a few interesting open problems.